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Fungrim entry: 0a3e5a

Symbol: ArgMin arg minP(x)f ⁣(x)\mathop{\operatorname{arg\,min}}\limits_{P\left(x\right)} f\!\left(x\right) Locations of minimum value
ArgMin(f(x), x, P(x)), rendered arg minP(x)f ⁣(x)\mathop{\operatorname{arg\,min}}\limits_{P\left(x\right)} f\!\left(x\right), gives the set of points rr satisfying P ⁣(r)P\!\left(r\right) such that f ⁣(r)=minP(x)f ⁣(x)f\!\left(r\right) = \mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right), if the minimum value exists.
If f ⁣(x)f\!\left(x\right) does not attain a minimum value on the set of points defined by P ⁣(x)P\!\left(x\right), the result is the empty set {}\left\{\right\}.
The argument x to this operator defines a locally bound variable. The corresponding predicate P ⁣(x)P\!\left(x\right) must define the domain of xx unambiguously; that is, it must include a statement such as xSx \in S where SS is a known set. More generally, x can be a collection of variables (x,y,)\left(x, y, \ldots\right) all of which become locally bound, with a corresponding predicate P ⁣(x,y,)P\!\left(x, y, \ldots\right).
Definitions:
Fungrim symbol Notation Short description
ArgMinarg minP(x)f ⁣(x)\mathop{\operatorname{arg\,min}}\limits_{P\left(x\right)} f\!\left(x\right) Locations of minimum value
MinimumminP(x)f ⁣(x)\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right) Minimum value of a set or function
Source code for this entry:
Entry(ID("0a3e5a"),
    SymbolDefinition(ArgMin, ArgMin(f(x), x, P(x)), "Locations of minimum value"),
    Description(SourceForm(ArgMin(f(x), x, P(x))), ", rendered", ArgMin(f(x), x, P(x)), ", gives the set of points", r, "satisfying", P(r), "such that", Equal(f(r), Minimum(f(x), x, P(x))), ", if the minimum value exists."),
    Description("If", f(x), "does not attain a minimum value on the set of points defined by", P(x), ", the result is the empty set", Set(), "."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC